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ARTICLESPUBLISHED ONLINE: 13 JUNE 2016 | DOI: 10.1038/NMAT4661

Crystal symmetry breaking and vacancies incolloidal lead chalcogenide quantum dotsFederica Bertolotti1, Dmitry N. Dirin2,3, Maria Ibáñez2,3, Frank Krumeich2, Antonio Cervellino4,Ruggero Frison5,6, Oleksandr Voznyy7, Edward H. Sargent7, Maksym V. Kovalenko2,3,Antonietta Guagliardi6* and Norberto Masciocchi1*

Size and shape tunability and low-cost solution processability make colloidal lead chalcogenide quantum dots (QDs) anemerging class of building blocks for innovative photovoltaic, thermoelectric and optoelectronic devices. Lead chalcogenideQDs are known to crystallize in the rock-salt structure, although with very dierent atomic order and stoichiometry in thecore and surface regions; however, there exists no convincing prior identification of how extreme downsizing and surface-induced ligand eects influence structural distortion. Using forefront X-ray scattering techniques and density functionaltheory calculations, here we have identified that, at sizes below 8nm, PbS and PbSe QDs undergo a lattice distortion withdisplacement of the Pb sublattice, driven by ligand-induced tensile strain. The resulting permanent electric dipoles may haveimplications on the oriented attachment of these QDs. Evidence is found for a Pb-deficient core and, in the as-synthesizedQDs, for a rhombic dodecahedral shape with nonpolar 110 facets. On varying the nature of the surface ligands, dierencesin lattice strains are found.

Colloidal semiconductor nanocrystals, also known as quantumdots (QDs) owing to their strong quantum-size effects, arean important class of building blocks for electronic, opto-

electronic and thermoelectric devices1,2. Owing to their exceptionalsize- and shape-tunable optical and electronic properties, as wellas their low-cost solution processability, lead chalcogenide (PbE,E = S, Se, Te) QDs have attracted a great deal of attention at anincreasing pace in the past 25 years, and a number of solid-state de-vice applications (such as diode lasers, infrared detectors and solarcells) are being pursued at present3–5. In PbE QDs, the electronicand optical characteristics are strongly influenced by the structureand stoichiometry of the nanocrystals, both at their interior andat their surface. These species crystallize in the highly symmetricrock-salt crystal structure; a metastable rhombohedral phase (ofthe α-GeTe type) has been theoretically predicted for bulk PbS6and temperature-dependent cation disordering has been reportedfor bulk PbS and PbTe, with no macroscopic crystal symmetrychange7,8. For nanometre-sized QDs, lattice and structural distor-tions induced by the extreme downsizing of the nanocrystals, bytheirmorphology and/or by surface-induced ligand effects, have notbeen detected so far. Concerning stoichiometry, colloidal synthesistypically provides metal-rich PbE QDs, with a larger metal excessfor smaller sizes and with anionic ligands preserving charge neutral-ity9,10. There exists a general consensus that nanocrystals possess astoichiometric core whereas the excess metal cations form a labilesurface layer. This external layer reversibly dissociates from, andbinds to, the crystal surface in the form of metal–ligand complexes,resulting in dynamic chemical formulae, surface reconstructionphenomena and QD size reduction over time11,12. All these effects

strongly influence the reactivity and optoelectronic properties ofthe QDs. Therefore, controlling the stoichiometry of the QDs isof utmost importance for understanding the relationship betweentheir properties and structure13. However, the facile adsorption anddesorption of metal complexes at the nanocrystal surface make theaccurate description of QD stoichiometry a challenging task forstandard analytical methods.

Using X-ray total scattering techniques on colloidal organicoleate(OA)-capped PbS and PbSe QD solutions and a modelbased on the Debye scattering equation (DSE)14, we experimen-tally unveil the distortion of the rock-salt structure towards a non-centrosymmetric rhombohedral (R3m) structure, due to the dis-placement of the Pb sublattice along the [111] axis (Fig. 1a). This dis-tortion is confirmed using density functional theory (DFT) calcula-tions and is probably triggered by the stereochemical effect of the 6s2lone pair of Pb(II) ions6, favoured by ligand-induced tensile stress inspherical nanoparticles. Our analysis also provides strong evidenceof an unexpected compositional picture of such QDs: a Pb-deficientcore and a Pb-enriched shell within a homo-core-shell model.

The rock-salt structure of PbE nanocrystals in the conventionalface-centred cubic lattice is shown in Fig. 1a; the face-centred cubiclattice is equally well described by the primitive rhombohedralunit cell emphasized by the shaded boundaries in Fig. 1a, andcharacterized by the interaxial angle αRH. This primitive unit cellis here used to describe both the pristine cubic lattice (αRH= 60)and the rhombohedrally distorted lattice resulting from a tensile(αRH<60) deformation along the [111] direction.

Highly monodisperse PbS and PbSe QDs were analysedby generating atomistic models of spherical homo-core-shell

1Dipartimento di Scienza e Alta Tecnologia and To.Sca.Lab, Università dell’Insubria, via Valleggio 11, I-22100 Como, Italy. 2Department of Chemistry andApplied Biosciences, ETH, Vladimir-Prelog-Weg 1, CH-8093 Zürich, Switzerland. 3Empa-Swiss Federal Laboratories for Materials Science and Technology,Überlandstrasse 129, CH-8600 Dübendorf, Switzerland. 4SLS, Laboratory for Synchrotron Radiation—Condensed Matter, Paul Scherrer Institut, CH-5232Villigen, Switzerland. 5Department of Chemistry, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. 6Istituto di Cristallografiaand To.Sca.Lab, CNR, via Valleggio 11, I-22100 Como, Italy. 7The Edward S. Rogers Department of Electrical and Computer Engineering, University ofToronto, 10 King’s College Road, Toronto, Ontario M5S 3G4, Canada. *e-mail: [email protected]; [email protected]

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ARTICLES NATUREMATERIALS DOI: 10.1038/NMAT4661

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Figure 1 | Lattice distortion of the rock-salt structure in PbS QDs. a, Rock-salt structure of PbS with the primitive rhombohedral unit cell (shaded area).Tensile deformations of the cubic structure along the [111] direction (arrows) induce a contraction of the rhombohedral angle αRH to values smaller than60. b, Experimental synchrotron X-ray diraction data of colloidal PbS QDs (black trace) in hexane (blue trace) with the Debye total scattering(texture-free) pattern model superimposed as a green line and the dierence curve (red trace), at the bottom of each plot. c, Reduced χ2 for the best fit ofthe DSE models to each set of data at dierent αRH angles, with the corresponding cell volume variation. d, χ2

rel versus1rPb, the oset of the Pb atom along[111], from the rock-salt position [χ2

rel=χ2 (1rPb)/χ2(0)], for the three dierent sizes: 7.7 nm (red), 5.0 nm (black) and 3.2 nm (blue). We notice that the

Pb oset follows the deviation of the αRH from 60 (the larger the shift, the larger the deviation). The inset in d shows the distortion of the PbS6 octahedraresulting in three shorter (2.91 Å) and three longer (3.05 Å) Pb-S distances breaking the centrosymmetry of the rock-salt structure. In a and in the inset ofd, colours of atomic species represent: black, lead; yellow, sulfur. In c and d, solid lines represent guides for the eye.

nanocrystals, with adjustable core diameter, shell thickness, latticeparameter and stoichiometry, and also by using an innovative DSEapproach14 to simulate the reciprocal space scattering pattern whileoptimizing the model parameters to fit the experimental data. ForPbS, the analysis was performed on both freshly prepared samplesand their aged counterparts after partial oxidation by storage atambient conditions.

With respect to the structural reordering in very small QDs,PbS and PbSe are found to be very similar; herein we focus on thecase of PbS, considering three different sizes (3.2, 5.0 and 7.7 nm).Outcomes for PbSe are provided in the Supplementary Information.

The main results concerning rhombohedrally distorted model(s)of as-synthesized PbS QDs are summarized in Fig. 1b–d. Bylowering the αRH angle from 60 to 58.6, we investigated competingstructural models for each of the three samples. The reducedχ 2 (goodness of fit) of each model is given in Fig. 1c as afunction of αRH and clearly indicates that the best agreement isobtained for αRH values<60 (Fig. 1b and Supplementary Figs 2–4).The rhombohedral lattice distortion (confirmed by conventional

Rietveld analysis, see Supplementary Fig. 30) varies in inverseproportionwith size. The shallowestminimum(at 59.1) is observedfor the smallest size (3.2 nm)QDs, with deeperminima (at 59.5 and59.7) for the 5.0 and 7.7 nm QDs, respectively, in agreement withthe smearing effect of the reduced nanocrystal coherent domains.

In our search for the driving force inducing such latticedistortion, we observed a similar size dependence for the cell volumevariation (Fig. 1c). The volume expansion of the unit cell at thesmallest size is fairly large (>1.1%) compared, for example, to thevolume increase in GeTe, a mere 0.8% for a temperature variationof 400 C (ref. 15). The inverse relationship between the cubic-equivalent lattice parameter (V 1/3

cell ) and QD diameter D (Fig. 2a)provides a V 1/3

cell slightly smaller than the value of bulk galena,when extrapolated to large sizes, and a negative surface tensionvalue (γ =−3.86 eVnm−2). According to a previous prediction6,the negative pressure induces a tensile stress, enabling the shift ofPb atoms along the [111] direction, (prompted by the incipientstereochemical activity of the Pb(II) 6s2 lone pair6), which causes thelattice distortion. Therefore, different Pb-sublattice displacements

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NATUREMATERIALS DOI: 10.1038/NMAT4661 ARTICLESPbS−OA model Dipole moments

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Figure 2 | Size-dependent lattice expansion, ligands-induced tensile stress and dipole moment. a, Dependence of the cubic-equivalent lattice parameter(Vcell)1/3 on the nanocrystal diameter (D) for PbS QDs. (Vcell)1/3 expands on decreasing the QD size (red points). The solid black curve is obtained byfitting the (Vcell)1/3 values with an inverse dependence law on the sample diameter: aQDs=a0 (1−Ω/D), whereΩ=4γ /3B depends on the surfacetension (γ ) and on the bulk modulus (B), and a0 is the lattice parameter for D→∞ (approaching the bulk value abulk). We obtained γ =−3.86 eV nm−2,indicating a negative surface tension responsible for the observed lattice expansion. b, Schematic of a tensile surface stress arising from inter-liganddistances larger than the spacing between 110 surface adsorption sites; the view shows a thin (001) cut, horizontal and vertical axes are [100] and [010],respectively. c, PbS rhombohedrally distorted structure viewed along [011], with the dipole vectors (red arrows) originating from the shift of the Pb ions(augmented for graphical clarity) and the loss of centrosymmetry. A net dipole moment µ= 190D is estimated for the 3.2 nm QDs. In b and c, colours ofatomic species represent: black, lead; yellow, sulfur; red, oxygen; grey, carbon; white, hydrogen.

were investigated at the optimal αRH angle and the relative change ofχ 2 is shown in Fig. 1d. The displacement observed in our PbSQDs isminimal compared to that predicted for the bulkmaterial6, althoughin line with other estimates7,8, and correlates well with the amountof lattice distortion. The Pb shift causes three shorter and threelonger Pb-S distances to arise (2.91Å and 3.05Å for the smallestQDs), providing a (3+ 3) coordination of Pb by S, and the lossof centrosymmetry in the structure. DFT calculations performedon bulk PbS confirmed such ferroelectric-type structural distortionunder the effect of a hydrostatic tensile stress. In nanocrystals, suchtensile stress could be induced by ligands whose packing density isnot commensurate with the underlying adsorption sites16 (Fig. 2b).Reproducing the ensuing structural distortion of the nanocrystalscomputationally is more difficult owing to the unknown surfacestructure and ligands organization (Supplementary Fig. 29).

An important consequence of such a ferroelectric-typestructural distortion is the formation of a permanent electricdipole along [111], which may play a plausible role in theobserved spontaneous linear aggregation of PbE QDs into wire-like structures in suspension and in dipole-driven co-alignmentin oriented attachment phenomena17,18. Calculations on theunrelaxed experimentally determined 3.2 nm PbS QD structurewith Pb-sublattice shift produce a large dipole moment µ≈190 D(Fig. 2c); estimates for spherical PbSe are plausible in line withexperimental reports18 (see the Supplementary Information). Thecurrent literature attributes the dipole-driven oriented attachmentof PbE QDs to asymmetries in Pb- and E-terminated polar111 facets17, despite that the presence of unpassivated E-richfacets conflicts with the established consensus that nanocrystalsare Pb-terminated. Therefore, characterizing the morphologyof these QDs becomes of greatest importance. PbS QDs havebeen reported to show cubic and cuboctahedral shapes, witha subtle interplay of 100 and 111 forms19. 110 facets arereported to appear at the initial stages of nanocrystal formationand, being highly reactive, to be preferentially consumed duringgrowth17,20 (the disposition of the three types of facets is givenin the inset of Fig. 3b). Unambiguously determining the shapeof QDs in this work by direct imaging remained challenging,as most nanocrystals appeared roundish. DSE modelling of thesmall angle X-ray scattering (SAXS) signal of PbS colloids (in

the Porod’s region, sensitive to the local interface roughness)provides evidence for a rhombic dodecahedral morphology in theas-synthesized suspensions (Fig. 3a and Supplementary Figs 11and 12). This morphology is further supported by X-ray powderdiffractometry on dried samples. Similarly to other reports forCu2O nanocrystals21, diffraction data (Fig. 3c,d and SupplementaryFig. 14) show a significantly enhanced intensity of the 220 peak. Thecubic harmonics description of textural effects in a conventionalRietveld fit provided the three-dimensional (3D) shapes of therelative orientation distribution function (ODF) (Fig. 3c,d insets);the results indicate a nonrandomness of the QD orientation withpreferential alignment of their (110) planes as parallel to thesubstrate. The implicit relation between the symmetry-relatedpreferred orientation poles (indicated by 12 arrows) and theunderlying rhombic dodecahedral nanocrystal shape is shown inFig. 3g. Selected high-angle annular dark-field (HAADF-STEM)(Fig. 4a,b) and phase contrast scanning transmission electronmicroscopy (PC-STEM) (Fig. 4c) images also suggest that the habitof nanoparticles can very closely approximate that of a rhombicdodecahedron. A recent high-resolution TEM (HRTEM) andFourier analysis study of the orientational relationship of oleicacid-stabilized PbS QDs forming 2D superlattices showed that thelarge majority of nanoparticles (>70%) possess a [110] orientation,although a truncated octahedral shape was proposed22. All thesefindings indicate that the rhombic dodecahedron is the dominantmorphology in as-synthesized PbS QDs; the nonpolar nature of110 forms dismisses the hypothesis of asymmetries in the facetingof nanocrystals, making the ferroelectric-type structural distortiona plausible cause for the formation of a dipolar moment.

On oxidation, a less pronounced ODF isosurface isfound (Fig. 3e,f), which is consistent with the presence ofrhombicuboctahedral clusters bound by polar 111 and nonpolar100 and 110 facets23 (appearing as red, blue and green colouredfacets, respectively, in Fig. 3h). The SAXS-DSE simulations (Fig. 3b)support such a morphological change. These findings suggest thatthe initially formed rhombic dodecahedral shape (possessing 110facets of relatively high surface tension24) progressively evolvesinto more stable forms25. Despite the fact that PbS QDs undergosurface reconstruction, theymaintain the original ferroelectric-typestructural distortion. The simultaneous appearance of 100 and



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Figure 3 | Morphology evolution of PbS QDs. a,b, SAXS signal (black trace) of freshly prepared (a) and aged (b) PbS 5.0 nm QDs, with the DSE simulation(red trace). The insets show the nanocrystal morphology: rhombic dodecahedral (exposing only 110 facets) and rhombicuboctahedral (exposing also111 and 100 facets), respectively. Colours of atomic species represent: black, lead; yellow, sulfur. c,d, X-ray powder diraction data of dried samplesmeasured in flat-plate geometry (black trace), showing the strong texture along the (220) planes for the freshly prepared 3.2 and 5.0 nm QDs. The greenline describes the Rietveld fit correcting the texture eects, the underlying blue line describes the simulated diraction trace of randomly oriented QDs.The 3D isosurfaces (insets), describing the orientation distribution functions, indicate the nonrandomness of the nanocrystal orientation, with their (110)planes parallel to the substrate. e,f, X-ray powder diraction data showing that the eect of preferred orientation fades on sample ageing with increasing111 and 200 peak intensities. g,h, Geometric relation between the preferred orientation poles and the underlying rhombic dodecahedral (g) orrhombicuboctahedral (h) shape. Black arrows represent the directions perpendicular to the nanocrystal facets parallel to the preferred orientation poles.

(asymmetrically distributed) polar 111 facets provide additionalweaker dipolar effects (Supplementary Fig. 25), which are howeverinsufficient to explain the experimentally reported values18,25.Considering the intricate interplay between the dipolar forces ofdifferent origin and other short-range interactions, it is not possibleto provide definitive proof that an internal dipole is the primarycausative agent in the observed 1D and 2D attachment of PbEnanocrystals of larger sizes and different morphology17,18,23,26.

However, whether the rhombohedral distortion in spherical-like OA-capped QDs and the Pb off-centring are further due tothe extreme downsizing of the PbE nanocrystals cannot be ruledout. Disentangling size from ligand-induced effects would require

studying QDs with much shorter carboxylates, or even uncappedones, which are unavailable as stable colloidswith reasonable controlover size and size distribution. Nevertheless, when we moved toinorganic ligands27 via solution exchange of the long-chain oleateswith short PbI3− or AsS43− fragments in PbS QDs, we observedmarkedly different effects on the X-ray scattering patterns (Fig. 5a).The splitting of h00 Bragg peaks in iodoplumbate-capped QDs isneither compatible with the undistorted cubic rock-salt structurenor with its rhombohedral distortion; therefore, a different kindof structural defectiveness, possibly driven by a more complexsurface rearrangement, must be considered28 (see SupplementaryInformation and Supplementary Figs 31 and 32). The sharpening of

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NATUREMATERIALS DOI: 10.1038/NMAT4661 ARTICLES

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Figure 4 | STEM analysis of the PbS QD structure and morphology. a,b, Spherical aberration-corrected HAADF-STEM image of the 7.7 nm sample (a)showing that the habit of the nanoparticles can very closely approximate a rhombic dodecahedron (b). Here the colours of atomic species represent: black,lead; yellow, sulfur. c, High-resolution phase contrast STEM image. d,e, Fourier transform of the highlighted regions in c shows the four-fold and two-foldsymmetry of crystals viewed along [001] (d) and [110] (e).

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Figure 5 | Organic-to-inorganic ligand exchange and induced lattice strain. a, Experimental synchrotron X-ray diraction data (subtracted of the solventsignal) collected on pristine OA-capped 3.5, 5.0 and 6.7 nm (blue, green, red traces) PbS colloids (top) and after ligand exchange with inorganic K3AsS4(intermediate) and MAPbI3 (bottom) ligands. The black rectangle highlights the highly reproducible splitting in MAPbI3-capped QDs of the 200 peak (lessevident, but fully observable also in the 400 one), which is compatible neither with the undistorted rock-salt structure nor with its rhombohedral distortion.The sharpening of the same h00 peaks, for both inorganic ligands, suggest the morphological elongation of nanocrystals parallel to the [100] direction. Thenarrow peaks in the MAPbI3 case cannot be attributed to contamination by the MAPbI3 phase (Bragg peak positions and intensities are shown forreference). Values in % refer to1a/abulk and indicate that the three ligands induce a dierent lattice strain. b,c, SAXS signal (black trace) of theK3AsS4-capped 3.5 nm (b) and 6.7 nm (c) PbS QDs, with the DSE simulation (green trace). The insets show the nanocrystal morphology: cubic-like(exposing extended 100 facets) in b and prismatic-like (originating from the fusion of two pristine clusters), in c. Colours of atomic species represent:black, lead; yellow, sulfur.



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curve (red lines) corresponds to a dierent stoichiometry model and Pb:S ratio (blue lines): PbSx model (0≤x≤ 1) with S vacancies throughout the entireQD volume; PbS/PbSx stoichiometric/non-stoichiometric core-shell model; and PbyS/PbSx core-shell model with a non-stoichiometric core (0≤ y≤ 1)and a non-stoichiometric shell (0≤ x≤ 1). b, Graphical view of PbSx (black, lead; yellow, sulfur), PbS/PbSx (black, lead; yellow, sulfur core; green, sulfurshell) and PbyS/PbSx (black, lead core; yellow, sulfur core; purple, lead shell; green, sulfur shell) atomistic models. c, Pb:S ratio and OA coverage of freshlyprepared (solid line) and aged (dashed line) QDs. d, QD overall diameter, resulting from the core size (orange) and the shell thickness (green) of thePbyS/PbSx model, for as-synthesized and oxidized samples. In a and c, solid lines represent guides for the eye.

the h00 peaks in both MAPbI3- and K3AsS4-capped QDs suggeststhat some morphological transformations also took place. Ontop of this, the two inorganic ligands provide markedly differentlattice strain (Fig. 5a and Supplementary Fig. 33). Thioarsenates,geometrically more adaptable to the QD surface, show tinyexpansions compared to the bulk lattice, with a negligible sizedependence, whereas iodoplumbates exhibit lattice expansions evenlarger than in oleates. These experimental observations indicatethat the nature of the ligands does indeed play a remarkable rolein modifying the lattice strain. Further analysis of the h00 peaksof the largest K3AsS4-capped PbS QDs suggests the formation ofelongated nanocrystals, the size along the [100] direction beingabout twice as large as the diameter of the pristine OA-cappedQDs. This finding suggests that they are formed by the fusion, insolution, through the 100 facets of two pristine QDs, after theirmorphological transformation occurring during ligand exchange.SAXS data confirm the evolution from multifaceted, spherical-like to more stable cubic-like nanocrystals (Fig. 5b,c) and thesubsequent oriented attachment before (or concomitantly to) thebinding of the new ligand. In such reconstruction, the dipole–dipoleinteraction of pristine QDs may bring the nanocrystals in close

proximity to each other, favouring the attachment after a minorreorientation29,30. In the smallest K3AsS4-capped PbS QDs thereis no evidence of fusion; however, the evolution towards a morestable cubic-like morphology, with extended 100 facets (Fig. 5b),is observed.

In our OA-capped PbS QDs, simultaneous with themorphological evolution, changes in QD composition, occurring atboth the surface and the interior of the nanocrystals, also emergedfrom the DSE analysis. The QD stoichiometry models used inthe DSE analysis are summarized in Fig. 6. A PbS/PbSx core-shellstructure with x ≤ 1 was initially considered, corresponding toa stoichiometric core and a S-deficient shell (Fig. 6c), followingthe most accepted model in other works9,10. Here, the Pb:S ratiodepends on the vacancies of S atoms in the shell. Considering themuch simpler spherical PbSx model in which S deficiency is allowedwithin the entire QD volume, and comparing the reduced χ 2 of thetwo models (plotted as a function of αRH in Fig. 6a), we find thatthe DSE method is sensitive to the region of the nanocrystal wherevacancies are located. When we allowed the Pb:S ratio to varyindependently within the core and shell (PbyS/PbSx , with x , y≤ 1,Fig. 6b), the agreement between the calculated and experimental

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NATUREMATERIALS DOI: 10.1038/NMAT4661 ARTICLESdata significantly improved for all samples, irrespective of the αRHangle. This finding indicates that the freshly prepared PbS QDshave a Pb-deficient core and a S-deficient shell, both showinga slight size dependence (Supplementary Table 5). Althoughnot totally unexpected (metal vacancies are reported in cationexchange mechanisms31), what is surprising in this work is the largeconcentration of Pb vacancies, which occur in the presence of otherstructural distortions. This finding closely resembles the case of theisostructural GeSbTe-based phase-change materials, stabilized bythe presence of up to 20% cation-site intrinsic defects32. Our DFTcalculations indicate a significantly reduced vacancy formationenergy in the rhombohedrally distorted lattice, because three outof six Pb-S bonds are weakened by the structural deformation.However, under tensile strain, the formation of up to∼10% vacantsites remains energetically favourable (Supplementary Fig. 26), wellbelow the DSE-based results. We attribute such overestimation tothe well-known anisotropic and anharmonic distribution of Pbatom displacements8,33, accounted for neither in powder diffractionanalyses nor in our model. Nevertheless, the χ 2 values for thevacancy level indicated by DFT remains markedly lower than for astoichiometric PbS core (see Supplementary Fig. 28).

Pb vacancies in the core may be driven by a number of concomi-tant factors related to the QD charge neutrality: the tendency ofexcess oleate to bind Pb atoms on the surface; the number of surface-bound oleates, which is limited by steric requirements; and the highsurface-to-volume ratio of the QDs. More importantly, photolumi-nescence quantum yields and DFT calculations demonstrate thatthe structural distortion and a limited amount of Pb vacancies arenot detrimental to the photophysical properties (see SupplementaryInformation for details and further discussion).

The relatively large content of S in the approximately 1-nm-thick QD shell and the relatively low Pb:S ratio of the QDschange significantly on sample oxidation. The results at the optimalαRH angle for each of the three QD sizes, for both pristine andaged QDs, are shown in Fig. 5c,d (see also Supplementary Fig. 7and Supplementary Table 5). In the pristine case, the calculatedtotal diameter agrees well with estimates from TEM analysis andabsorbance spectra (Supplementary Figs 17 and 19). At largersizes, the S-deficient shell thickens, resulting in slightly lower Pb:Sratios. On oxidation, samples undergo significant modificationsculminating in the formation of a Pb-terminated layer, larger Pb:Sratios (Fig. 6c, dashed green line, and Supplementary Table 5) andslightly smaller overall sizes (Fig. 6d and Supplementary Table 4)10.The concomitant depletion of Pb in the core and QD size reductioncould be attributed to a metal ion migration towards the surface,a vacancy-mediated process invoked in many experimental studiesof cation exchange31. Assuming that the number of oleates at theQD surface is solely determined by a charge balance mechanism,an average coverage of ∼2 and ∼4 oleates nm−2 is calculated, forfresh and oxidized samples, respectively (Fig. 6c, brown solid anddashed lines), in line with the experimentally determined valuesfor other syntheses9. The presence of additional neutral oleic acidmolecules or the action of other anionic ligands, which are likely tooccur during oxidative post-synthesis changes at theQD surface11,12,cannot be excluded.

The results presented here shed new light onto the crystalstructure of PbE QDs and reshape the understanding of theirstructural, compositional and morphological properties. Theseinsights highlight the intrinsic complexity of deceptively simplerock-salt structures on extreme downsizing and surface ligand-induced effects, and point to the importance of such studies for thedevelopment of QD-based devices.

MethodsMethods and any associated references are available in the onlineversion of the paper.

Received 16 June 2015; accepted 11 May 2016;published online 13 June 2016

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AcknowledgementsF.B. acknowledges University of Insubria for Junior Fellowship Grant 2013, M.V.K.acknowledges the European Union for financial support via FP7 ERC Starting Grant2012 (Project NANOSOLID, GA No. 306733), D.N.D. thanks the European Union forMarie Curie Fellowship (PIIF-GA-2012-330524) and M.I. thanks AGAUR for her Beatriui Pinós post-doctoral grant (2013 BP-A 00344). Synchrotron XRPD data were collected atthe X04SA-MS Beamline of the Swiss Light Source. M. Döbeli is gratefully acknowledgedfor taking RBS spectra. Electron microscopy was performed at the Scientific Center for

Optical and Electron Microscopy (ScopeM) at ETH Zürich. Computations wereperformed using the BlueGene/Q supercomputer at the SciNet HPC Consortiumprovided through the Southern Ontario Smart Computing Innovation Platform(SOSCIP). We thank N. Stadie and J. Mason for reading the manuscript.

Author contributionsA.G., N.M. and M.V.K. formulated the project. D.N.D. and M.I. synthesized thecompounds and performed the optical properties characterization. A.C., F.B., R.F., A.G.and N.M. collected and analysed the X-ray total scattering data. F.K. collected andanalysed the electron microscopy images. O.V. and E.H.S. performed DFT calculations.A.G. and N.M. wrote the manuscript, with the contribution of all authors.

Additional informationSupplementary information is available in the online version of the paper. Reprints andpermissions information is available online at www.nature.com/reprints.Correspondence and requests for materials should be addressed to A.G. or N.M.

Competing financial interestsThe authors declare no competing financial interests.

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NATUREMATERIALS DOI: 10.1038/NMAT4661 ARTICLESMethodsQD synthesis.We synthesized highly monodisperse PbS and PbSe QDs accordingto literature methods, with slight modifications. Details are provided in theSupplementary Information and Supplementary Table 1. Briefly, PbS and PbSeQDs were synthesized by reacting lead acetate and oleic acid, followed by additionof bis(trimethylsilyl)sulfide and tri(n-butyl)phosphineselenide, respectively.

QD characterization. The size of the PbS and PbSe QDs was determined bycombining three different techniques: transmission electron microscopy (TEM),absorption spectroscopy and wide angle X-ray total scattering (WAXS) techniques(through the DSE modelling). TEM images were collected on a FEI Tecnai F30microscope. Scanning transmission electron microscopy (STEM) images werecollected on an aberration-corrected Hitachi HD-2700CS instrument (Fig. 4 andSupplementary Figs 16–17). Visible–near infrared absorption spectra for dilutedcolloidal solutions in tetrachloroethylene were collected using a Jasco V670spectrometer, in the range 400–2,000 nm (for PbS) and 1,000–2,000 nm (for PbSe)(Supplementary Figs 19–21). Photoluminescence (PL) spectra and PL quantumyield (QY) measurements were performed on Fluorolog iHR 320 Horiba JobinYvon spectrofluorimeter equipped with a InGaAs detector (Supplementary Fig. 22).For PL QY measurements, QDs were excited at 960 nm by a 450W Xenon lamp andusing an IR-26 dye molecule as a standard.

Synchrotron X-ray scattering data were collected at the X04SA-MS PowderDiffraction Beamline of the Swiss Light Source of the Paul Scherrer Institut with22 and 25 keV beam energies in the 0.5–130 2θ range (Fig. 1b and SupplementaryFigs 7 and 8a).

The composition of the PbS and PbSe QDs was determined by DSE methods(Fig. 5a and c and Supplementary Figs 9b,c and Supplementary Tables 2 and 3) and,for PbS samples only, also by Rutherford backscattering spectrometry (RBS)performed at the ETH laboratory of Ion-Beam Physics (Supplementary Fig. 23 andSupplementary Table 2). The occurrence of crystalline contaminants prevented ameaningful RBS analysis of the PbSe QDs.

Textural effects on PbS and PbSe dried samples were determined usinglaboratory X-ray powder diffractometry (Fig. 3c–f and Supplementary Figs 13–15).Powder diffraction traces were collected using Cu Kα radiation (λ=1.5418Å) on aBruker AXS D8 Advance diffractometer, on samples deposited on a siliconmonocrystal zero-background sample holder. Following the conventional rules,hkl , (hkl), [hkl] and hkl notations are used in the main text and figures toindicate: reflection indices, crystallographic planes, directions and sets of facets (orforms), respectively (in the cubic setting). Further details on characterizationmethods are provided in the Supplementary Information.

DSEmethod. The DSE method was applied in this work to characterize very smallPbS and PbSe nanocrystals (size range 3–8 nm) in colloidal solutions. Debye’sformula describes the average differential cross-section of a randomly orientedcollection of particles, regardless of their ordered or disordered atomicarrangement. Compared to conventional diffraction techniques, the DSE takesadvantage of simultaneously modelling Bragg and diffuse scattering as afunction of the distribution of interatomic distances within the nanoparticle,as follows:

I(Q)=N∑j=1

fj(Q)2o2j +2N∑

j>i=1

fj(Q)fi(Q)Tj(Q)Ti(Q)ojoisin(Qd ij

)(Qd ij

) (1)

where Q=2πq, q=2sinθ/λ is the length of the reciprocal scattering vector, λ isthe radiation wavelength, fj is the atomic form factor, dij is the interatomic distancebetween atoms i and j, and N is the number of atoms in the nanoparticle. In thisformula, thermal vibrations (or static disorder) and partial occupancy factors arealso taken into account by two atomic, adjustable parameters (T and o). The firstsummation in equation (1) addresses the contribution of the (zero) distance of eachatom from itself, the second summation (the interference term) that of thenon-zero distances between pairs of distinct atoms.

The approach we used in this work is based on a new and fast implementationof the Debye equation, which makes use of sampled interatomic distances insteadof the original ones, to speed up calculations, as adopted by the Debussy Suite ofprograms14. The DSE method here was applied to determine structure, size, sizedistribution and stoichiometry of PbS and PbSe QDs from the WAXS patterncollected on their colloidal solutions. Simulations of SAXS signals in the Porod’sregion (which, owing to the small size of QDs involved, was directly accessible aspart of the WAXS pattern) allowed the morphology of the QDs to be inferred.Graphical outcome of the analysis and main results are shown in Figs 1b–d and 6and Supplementary Fig. 7 (PbS), Supplementary Figs 8 and 9 (PbSe),Supplementary Fig. 18 and Supplementary Tables 2–7. DSE-derived sizes and sizedistributions of both PbS and PbSe QDs were successfully used to calculateabsorption spectra in good agreement with the experimental ones, as shown inSupplementary Fig. 20 and detailed in the Supplementary Information.

The outcome of the analysis in the SAXS region is shown in Fig. 3a,b andSupplementary Figs 11 and 12. Briefly, in the WAXS region, the DSE-based methodincluded generating a population of core-shell nanocrystals in real space for eachαRH angle; using the set of interatomic distances in the Debye equation to simulatethe reciprocal space scattering pattern; and optimizing the model parameters to fitthe experimental data. Aiming at possibly differentiating the inner and the outerportions of the nanocrystals, refinement included the core diameter, the shellthickness, the site occupancy factor of Pbcore and Sshell atoms, along with the celledge. Details of the DSE method and the specific core-shell modelling are providedin the Supplementary Information. The observation of isotropic QDs byhigh-resolution transmission electron microscopy validated this choice of sphericalmorphology. In this work, we neglected the outer organic oleate (OA) layer whichis nearly invisible to X-rays.

Ab initio calculations. DFT simulations were performed using the QuickStepmodule of the CP2K program suite34. Generalized gradients approximation withPerdew–Burke–Ernzerhof functional, molecules-optimized double ζ -valencepolarized basis set and Goedecker–Teter–Hutter pseudopotentials at 300 Ry densitycutoff were used. A 30×30×30A3 unit cell with a single k-point was used for bulkPbS, and a 50×50×50A3 cell for∼3 nm nanocrystals. All structures were fullyrelaxed. The dipole moment of the distorted structures was calculated withoutperforming geometry optimization.

Code availability. Debussy Suite is freely available athttp://debussy.sourceforge.net.

References34. VandeVondele, J. et al . Quickstep: fast and accurate density functional

calculations using a mixed Gaussian and plane waves approach. Comput. Phys.Commun. 167, 103–128 (2005).

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